Theoretical Principles & Computational Physics Guide
Miscellaneous Computational Problems: Numerical Sequences & Special Algorithms
Dr. Alorika Chatterjee & Dr. Anirban Shaw
Science & Physics Learners
Python 3 / NumPy / SciPy
1. Core Principles & Theoretical Framework
Fundamental Recurrences: Explores famous numerical series including Fibonacci sequences, Pascal's triangle, and digital invariants.
Dynamic Programming: Demonstrates state memoization to transform exponential recursive algorithms into linear time.
Physical Analogies: Fibonacci ratios converge to the Golden Ratio $\phi = (1 + \sqrt{5})/2$, modeling phyllotaxis and quasi-crystal structures.
2. Mathematical Formulation & Algorithmic Pipeline
Fibonacci recurrence and convergence to the Golden Ratio:
$$ F_n = F_{n-1} + F_{n-2}, \quad \lim_{n\to\infty} \frac{F_n}{F_{n-1}} = \phi = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887 $$
3. Numerical Mechanics & Implementation Logic
- Iterative State Shift: Maintain two variables $(a, b)$; update in parallel via
a, b = b, a + b. - Memoization: Cache computed sequence values to avoid redundant tree evaluation.
- Bitwise Checks: Use modular arithmetic and digital decomposition to check numerical symmetry properties.
4. Relevant Computational Practice Exercises
Exercise 1
Generate the first 50 Fibonacci numbers iteratively and plot the ratio $F_n / F_{n-1}$ demonstrating convergence to $\phi$.
Exercise 2
Write a function to check if a 3-digit number is an Armstrong number (sum of cubes of digits equals the number).
Exercise 3
Generate Pascal's Triangle up to $N = 10$ rows and show how each row corresponds to binomial coefficients $\binom{n}{k}$.
- Donald E. Knuth, "The Art of Computer Programming", Addison-Wesley (1997).